2018 AMC 10B Problems
Contents
- 1 Problem 1
- 2 Problem 2
- 3 Problem 3
- 4 Problem 4
- 5 Problem 5
- 6 Problem 6
- 7 Problem 7
- 8 Problem 8
- 9 Problem 9
- 10 Problem 10
- 11 Problem 11
- 12 Problem 12
- 13 Problem 13
- 14 Problem 14
- 15 Problem 15
- 16 Problem 16
- 17 Problem 17
- 18 Problem 18
- 19 Problem 19
- 20 Problem 20
- 21 Problem 21
- 22 Problem 22
- 23 Problem 23
- 24 Problem 24
- 25 Problem 25
- 26 See also
Problem 1
Kate bakes a -inch by
-inch pan of cornbread. The cornbread is cut into pieces that measure
inches by
inches. How many pieces of cornbread does the pan contain?
Problem 2
Sam drove miles in
minutes. His average speed during the first
minutes was
mph (miles per hour), and his average speed during the second
minutes was
mph. What was his average speed, in mph, during the last
minutes?
Problem 3
In the expression each blank is to be filled in with one of the digits
or
with each digit being used once. How many different values can be obtained?
Problem 4
A three-dimensional rectangular box with dimensions ,
, and
has faces whose surface areas are
and
square units. What is
?
Problem 5
How many subsets of contain at least one prime number?
Problem 6
A box contains chips, numbered
and
. Chips are drawn randomly one at a time without replacement until the sum of the values drawn exceeds
. What is the probability that
draws are required?
Problem 7
In the figure below, congruent semicircles are drawn along a diameter of a large semicircle, with their diameters covering the diameter of the large semicircle with no overlap. Let
be the combined area of the small semicircles and
be the area of the region inside the large semicircle but outside the small semicircles. The ratio
is
. What is
?
Problem 8
Sara makes a staircase out of toothpicks as shown:
This is a -step staircase and uses
toothpicks. How many steps would be in a staircase that used
toothpicks?
Problem 9
The faces of each of standard dice are labeled with the integers from
to
. Let
be the probability that when all
dice are rolled, the sum of the numbers on the top faces is
. What other sum occurs with the same probability
?
Problem 10
In the rectangular parallelepiped shown, ,
, and
. Point
is the midpoint of
. What is the volume of the rectangular pyramid with base
and apex
?
Problem 11
Which of the following expressions is never a prime number when is a prime number?
Problem 12
Line segment is a diameter of a circle with
. Point
, not equal to
or
, lies on the circle. As point
moves around the circle, the centroid (center of mass) of
traces out a closed curve missing two points. To the nearest positive integer, what is the area of the region bounded by this curve?
Problem 13
How many of the first numbers in the sequence
are divisible by
?
Problem 14
A list of positive integers has a unique mode, which occurs exactly
times. What is the least number of distinct values that can occur in the list?
Problem 15
A closed box with a square base is to be wrapped with a square sheet of wrapping paper. The box is centered on the wrapping paper with the vertices of the base lying on the midlines of the square sheet of paper, as shown in the figure on the left. The four corners of the wrapping paper are to be folded up over the sides and brought together to meet at the center of the top of the box, point in the figure on the right. The box has base length
and height
. What is the area of the sheet of wrapping paper?
Problem 16
Let be a strictly increasing sequence of positive integers such that
What is the remainder when
is divided by
?
Problem 17
In rectangle ,
and
. Points
and
lie on
, points
and
lie on
, points
and
lie on
, and points
and
lie on
so that
and the convex octagon
is equilateral. The length of a side of this octagon can be expressed in the form
, where
,
, and
are integers and
is not divisible by the square of any prime. What is
?
Problem 18
Three young brother-sister pairs from different families need to take a trip in a van. These six children will occupy the second and third rows in the van, each of which has three seats. To avoid disruptions, siblings may not sit right next to each other in the same row, and no child may sit directly in front of his or her sibling. How many seating arrangements are possible for this trip?
Problem 19
Joey and Chloe and their daughter Zoe all have the same birthday. Joey is year older than Chloe, and Zoe is exactly
year old today. Today is the first of the
birthdays on which Chloe's age will be an integral multiple of Zoe's age. What will be the sum of the two digits of Joey's age the next time his age is a multiple of Zoe's age?
Problem 20
A function is defined recursively by
and
for all integers
. What is
?
Problem 21
Mary chose an even -digit number
. She wrote down all the divisors of
in increasing order from left to right:
. At some moment Mary wrote
as a divisor of
. What is the smallest possible value of the next divisor written to the right of
?
Problem 22
Real numbers and
are chosen independently and uniformly at random from the interval
. Which of the following numbers is closest to the probability that
and
are the side lengths of an obtuse triangle?
Problem 23
How many ordered pairs of positive integers satisfy the equation
where
denotes the greatest common divisor of
and
, and
denotes their least common multiple?
Problem 24
Let be a regular hexagon with side length
. Denote by
,
, and
the midpoints of sides
,
, and
, respectively. What is the area of the convex hexagon whose interior is the intersection of the interiors of
and
?
Problem 25
Let denote the greatest integer less than or equal to
. How many real numbers
satisfy the equation
?
See also
2018 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by 2018 AMC 10A |
Followed by 2019 AMC 10A | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
All AMC 10 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.